Role of quasiparticles in the growth of a trapped Bose-Einstein condensate

4 Pages·0.15 MB·English

by  C. W. Gardiner

Role of quasiparticles in the growth of a trapped Bose-Einstein condensate
C.W. Gardiner1 M.D. Lee1 R.J. Ballagh2 M.J. Davis2 and P. Zoller3.
1 School of Chemical and Physical Sciences, Victoria University, Wellington, New Zealand
2 Physics Department, University of Otago, Dunedin, New Zealand
3 Institut fu¨r theoretische Physik, Universit¨at Innsbruck, A6020 Innsbruck, Austria.
Amajorextensionofthemodelofcondensategrowthintroducedbyusismadetotakeaccountof
theevolution of theoccupations oflower traplevels (quasiparticles) byscattering processes, and of
thefullBose-Einsteinformulafortheoccupationsofhighertraplevels,whichareassumedtohavea
timeindependentoccupation. Theprincipaleffectisaspeedupofthegrowthratebysomewhatless
thananorderofmagnitude,theprecisevaluedependingonthetheassumptionsmadeonscattering
and transition rates for thequasiparticle levels.
8
9
9
1 Inapreviouspaper[1],weintroducedaformulaforthe {exp[(E −µ)/k T]−1}−1. The value of E will be
B max
n growth of a Bose-Einstein condensate, in which growth assumedtobe smallenoughforthe majorityofatomsto
a resulted exclusively from stimulated collisions of atoms have energies higher than E , so that this part of the
J max
where one of the atoms is left in the condensate. This bath can be treated as being essentially undepleted by
5
gave a very simple growth formula which predicts a rate the process of condensate growth.

of growth of the order of magnitude of that observed
1
v in current experiments [2–4]. The direct stimulated ef-
7 fect must eventually be very important, once a signifi-
2 cantamount ofcondensate has formed, but in the initial
0 stagestherewillofcoursealsobeasignificantnumberof 0
1
transitionstoexcitedstatesofthe condensate(quasipar-
0
8 ticles),whosepopulationswillthenalsogrow. Aswellas
9 this, there will be interactions between the condensate,
/ the quasiparticles and the atomic vapor from which the
t 0
a
condensate forms. This paper extends the description of
m
the condensate growth to include these factors. Fig.1: Modification of the equilibrium level occupation by
- condensate growth, for a 3D harmonic trap. The cumulative
d All of these effects are encompassed by the descrip-
occupationofstatesN(E)withenergybelowE,isshownfor
n tion given in [1,5]. However, the practical extension of
o this description to take account of the additional effects agasofnoninteractingatomsasadottedline,andaftercon-
c densate growth (when the ground level energy has increased
would involve a calculation of all the eigenfunctions for
v: thetrappedcondensate,andthedetailedsummationover by µ(N)) is shown as a dashed line. The solid line gives
i the occupation f(E) = dN(E) after condensate growth. The
X all processes involving these. Since the number of levels dE
values of µ,µ(N), and kBT correspond to the MIT sodium
involved is of the order of tens of thousands, this could
r condensate of 106 atoms at 1.2 µK.
a be a formidable task. However some quite reasonable
estimates can be made for the overall effects of these
The energy levels between E and E are taken to
processes, and from these we can derive a set of easily R max
haveatimedependentpopulation,sincethecontinuation
solvable differential equations for the populations of the
of the equilibrium Bose-Einstein formula to lower ener-
condensate and the lower energy quasiparticles.
gies eventually leads to an unrealistic situation in which
Asinourpreviouswork,wedividethestatesinthepo-
the populations and transition rates become too large
tential into the condensate band, RC, which consists of for the populations to be considered to be constant—
the energy levels significantly affected by the presence of
this demonstrates that the initial condition in which the
acondensateinthegroundstate,andthenon-condensate vapor is at a positive chemical potential cannot apply
band, R , which contains all the remaining energy lev-
NC for all energies. The choice of E is thus determined
max
elsabovethecondensateband. Thedivisionbetweenthe
as a lower limit to the equilibrium distribution, with the
two bands is taken to be at the value, ER. distribution in the range between Emax and ER treated
The picture we shall use assumes that R consists as time dependent, and computed as part of the growth
NC
of a large “bath” of atomic vapor, whose distribution kinetics.
function is given, for the energy levels greater than a The value E , above which the energy levels are
R
value which we shall call E (with E > E ), by a taken as unperturbed, was fixed at 2µ. This value and
max max R
time-independent equilibrium Bose-Einsteindistribution the ground state energy level—the chemical potential
1
µ(N)—put bounds on the energy levels of the states in + n˙+ −n˙− . (4)
m m
between. As a simple expression of this fact, the levels m
X(cid:8) (cid:9)
between µ(N) and E are determined by interpolating
R
linearly between the two extremes, using a density of Here 2(N +1)W+(N) is the transition rate for an atom
statesN[E−µ(N)]2, whereN is anormalizationchosen to enter the condensate level as a result of a collision
sothatthecumulativenumberofstatesmatchesthecor- between two atoms in the “bath” of atomic vapor—the
responding cumulative value for an unperturbed three factor N +1, takes account of both the “spontaneous”
dimensional harmonic oscillator when E = E . As an term, and the “stimulated” term induced by the pres-
R
illustration of the effect of this, we show in Fig.1 the cu- enceofthecondensate. Thereversedprocessoccurswith
mulative occupation N(E), and the occupation per unit the overall rate 2NW+(N)e(µ(N)−µ)/kBT—that is with
energy interval f(E) = dN(E)/dE, when the system is no “spontaneous” term, and with a factor dependent on
in equilibrium. (The condensate population itself is not the difference of the chemical potential µ of the vapor,
shown.) and that µ(N) of the condensate. As a result, equilib-
rium in the large N limit occurs at equality of the two
Bath of higher energy atoms Bath of higher energy atoms
chemical potentials.
q
p m p m During the process of BEC formation, the spectrum
Emax Emax
n q n of eigenvalues makes a transition from the unperturbed
spectrumoftraplevelstothecasewherethespectrumis
strongly affected by the condensate in the ground state.
The Bogoliubov spectrum of a condensed gas is valid in
the case where the number of particles in the conden-
Emax Emax
sate, n , is so large that it is valid to write n ≈ N.
0 0
Thus, during the initial stages of condensate formation,
where this is not true, one must use another formalism.
In this paper we will consider the situation in which the
interaction between the particles is very weak, as is in
Fig.2: The transitions being considered: Left—scattering; practicethe case. Thismeansthatwe willbe abletouse
Right—Condensate growth. the unperturbed spectrum for the initial stages of con-
densation, and only use the Bogoliubov description once
Thedynamicswewillconsiderwillarisefromtwokinds enough condensate has formed to make the effective in-
of process as illustrated in Fig.2. teraction rather stronger.
Growth: A collision between a pair of atoms initially in
The basic formalism of [5] can still be carried out in
thebathofatomicvaporresultsinoneoftheatomshav-
this case, and the modification that is found is rather
ing a final energy less than E .
max minor—essentially, we make the substitution N →n in
0
Scattering: A collision between an atom initially in an the chemical potential and the W+(N),W++(N) func-
energy level below Emax and a bath atom transfers the tions, and set W−+ → 0, since this term comes from
m
first atom to another energy level below E .
max the mixing of creation and annihilation operators which
Our treatment therefore omits any scattering between
arises from the Bogoliubov method. In order to simplify
atoms whichbothhaveenergiesless than E ,which is
max the equations we also group the levels in narrow bands
reasonable if the number of atoms in the bath is almost
ofenergywith g levels per group,andfor simplicity use
k
100% of the total number of atoms.
the same notation n now for the number of particles in
k
The mechanism of condensate growth, as in [1,5], un-
theenergybandwithmeanenergye . (Thiscorresponds
k
derappropriateapproximations,givesrisetoequationsof
to the ergodic assumption used in [6].) We then deduce
motion for the number of atoms in the condensate band
N,andthe numberofquasiparticleexcitationsn (with
energies em < Emax) in the condensate, whichmcan be n˙m|growth ≡2Wm++(n0) 1−eµ(n0k)B−µT+em nm+gm ,
written as follows. First define (cid:26)(cid:20) (cid:21) (cid:27)
(5)
n˙+m ≡2Wm++(N) (1−e(µ(N)−µ+em)/kBT)nm+1 , (1) n˙0|growth =2W+(n0) 1−eµ(knB0)T−µ n0+1 . (6)
n˙− ≡2W−+(N)h(1−e(−µ(N)+µ+em)/kBT)n +i1 . (2) (cid:26)(cid:20) (cid:21) (cid:27)
m m m
h i The growth equations can be modified to include the
Then the multilevel growth equations are
terms derived in [5] corresponding to the scattering of
n˙ =n˙+ +n˙−, (3) particles in the condensate band by the vapor particles.
m m m
This is equivalent to scattering of particles by a heat
N˙ =2W+(N) (1−e(µ(N)−µ)/kBT)N +1 bath, whichleads to a rate equationfor scatteringof the
(cid:16) (cid:17)
2
form(WhereN¯ =1/(exp[(e −e )/k T]−1),andby used were chosen so as to be in approximate agreement
km k m B
k >m we mean e >e ) with the experimental work being conducted at MIT,
k m
wherethegrowthofBose-Einsteincondensatesof23Nais
n˙m|scatt = being studied. Incontrasttothe estimatein [1]inwhich
γ N¯ n (n +g )−(N¯ +1)(n +g )n the bath distribution was approximated by a Maxwell-
mk mk k m m mk k k m
Boltzmann distribution, in this computation we use the
k<m
X (cid:8) (cid:9) full Bose-Einstein distribution, truncated at a lower en-
+ γ (N¯ +1)n (n +g )−N¯ (n +g )n .
km km k m m km k k m ergy of E , since lower energies are described by the
max
kX>m (cid:8) (cid:9) nm variables.
(7)
In applying the theory two major approximations are
made. Firstly, the W++(N) functions were assumed
The formulaeofQKIII giveprecisemethods for comput- m
to be equal to the W+(N) function, since the actual
ing the coefficients γ , but we can simplify their com-
km W++(N) terms are not easily calculated. The justifi-
putation by adapting the kinetic equation of Holland et m
cation for this is that the W++(N) terms represent an
al. [6]. This methodology is based on a model in which m
averagingoverall the levels contained in the mth group,
the trap levels are all treated as being unaffected by the
and as such they may be expected to be of the same or-
presenceofthecondensate,whichshouldsufficeasafirst
der of magnitude as W+(N). As a validity check, it was
approximation. To apply it to this situation, we assume
found that the effect on the condensate growth rate was
all the levels with energies greater than E are ther-
max small when the W++(N) terms were altered by a factor
malized, and sum out over these levels. The working is m
in the range 0.5−2.
essentiallystraightforward,andyieldsanequationforthe
In current BEC experiments the confining harmonic
n variables in the form (with M the mass of the atom
m potential is normally anisotropic, whereas [6] was re-
and a the scattering length)
stricted to an isotropic trap. The second approximation
8Ma2ω2 is therefore that the scattering rate factor Γ(T) is equal
n˙m|scatt = π¯h eµ/kBTΓ(T)× to that for an isotropic 3D harmonic oscillator with fre-
quency equal to the geometric mean frequency of the
1
n (g +n )e−h¯ωmk/kBT −n (g +n ) anisotropic trap. The precise value of this factor was
k m m m k k
( gm found to have little effect on the solutions, so long as it
kX<m h i
was greater than about one tenth of the value given by
1
+ n (g +n )−n (g +n )e−h¯ωkm/kBT . (9).
k m m m k k
g
k ) Thecondensaterateofgrowthdependsonthenumber
kX>m h i
(8) of groups of levels considered in the model—modeling
moregroupsinthe condensatebandincreasestherateof
where Γ(T) ≡ e−em/kBT has a value which growth, which approaches a limiting value. The number
em>emax
depends on the spectrum of energies. For an isotropic of groups modeled was therefore chosen as large as pos-
P
3-dimensional harmonic oscillator with frequency ω, the sible,butitwasrequiredthattherewereatleast4levels
energy levels above the zero point are e =n¯hω, so that in the first group of levels above the condensate level.
n
we find
a) b)
Γ(T)= e−Emax/kBT . (9) ber106 ber106
1−e−h¯ω/kBT um um
N N
This corresponds to essentially to (7) when one makes nsate nsate
the correspondences nde 0.5 m K nde 1.0 m K
o o
C 0 C 0
N¯km →e−(ek−em)/kBT, 1+N¯km →1 (10) 0 Time in 1seconds 2 0 Time in0 s.5econds 1
8Ma2ω2eµ/kBTΓ(T)
γ =γ → with k >m. (11) Fig.3: Condensate growth for Sodium: Dotted line—Growth
km mk
π¯h g
k of the condensate for the uncorrected model of [1]; Dashed
line—scattering is neglected, but W+ is given by using the
The equation for both growth and scattering is now
Bose-Einstein correction; Solid line—with scattering and the
given by adding (5) to (10)
Bose-Einstein correction. In all cases the initial amount of
n˙ =n˙ | +n˙ | , (12) condensate at t=0 was 500 atoms.
m m growth m scatt
The initial populations for the groups in the condensate
where, for m=0, we use (6) instead of (5). band were generated by a method which models the ex-
The overall evolution of the system can now be found perimental procedure. We start at some initial t ≪ 0
from the numerical solutions to (12). The parameters with the bath at a chemical potential µ ≈ 0, and evolve
3
the equations of motion until equilibrium is reached; at is different, since Stoof’s Fokker-Planck equation treats
t = 0 we then change µ to a positive value such that the untrapped case, where the low-lying levels are ex-
µ = µ(N ), where N is the final number of atoms tremely closely spaced, and relative coherences between
final final
in the condensate. Changing the populations at t = 0 these levelsbecome important. The currentexperiments
merely moves the growthcurve slightly forwardor back- are probably closer to the situation of our model, since
ward in time, with no effect on its basic shape. We the growthrate is somewhatslower than the lowest trap
present a sample of the results obtained in Fig.3. The frequency.
trap parameters, νx = 18.5Hz,νy = νz = 250Hz, tem- We can conclude from the results of this model that
peratures andfinal condensate number are chosenin the the main effect of the inclusion of the scattering and the
range presently being investigated for sodium. computationof W+ using the full Bose-Einsteinformula
The typical behavior of the noncondensate levels is is to speed up the condensate growth by up to one or-
shown in Fig.4, for the case where the initial occupa- der of magnitude, the precise speedup depending on the
tion of all groups below ER is chosen to be zero, in or- condensate size and temperature. Even though we have
der to show the speed of the relaxation process. The onlyestimatedthescatteringandtransitionratesforthe
initial growth is in the population of the noncondensate quasiparticlelevels,theresultsarereasonablypredictive,
levels—theiroccupationscanbecomeverylarge,butthis sinceitwouldbehardtocreditthescatteringortheW++
is because the numbers of levels in each group are very factors as being very different from the values assumed,
large, so that there is little degeneracy, i.e., the number and also because the numerical predictions are not very
ofatomsperleveldoesnotsignificantlyexceedone. Even sensitive to these precise values. Precise predictions will
inthelowestgroupthedegeneracyisnomorethatabout involve the detailed computation of these effects, rather
100. The moment the condensate achieves a significant than their estimation.
degeneracy,thestimulatedprocesstakesover,andimme-
Acknowledgments: We would like to thank Wolfgang
diatelydrawstheexcesspopulationofthenoncondensate
Ketterle and Hans-Joachim Miesner for discussions re-
groups into the condensate, well before the full conden-
gardingsodiumexperiments,andEricCornell,CarlWie-
sate occupation is achieved. Thus, apart from the initial
man, Deborah Jin and Jason Ensher for discussions re-
transient, the condensate growth occurs essentially by
gardingrubidiumexperiments. Thisworkwassupported
thesamemechanismasin[1],withthemodificationthat
by the Marsden Fund under contract number PVT-603,
the distribution over the noncondensate levels changes
and by O¨sterreichische Fonds zur F¨orderung der wis-
slowlyinresponsetothechangeofthecondensatechem-
senschaftlichen Forschung.
ical potential.
Ifscatteringisentirelyneglectedthepopulationsofthe
lowest noncondensate levels become several times larger
than those of the the condensate before settling to their
very much lower equilibrium values. However, the inclu-
sionofevenaslittle as0.1%ofthe strengthofscattering
used here eliminates that effect almost entirely. [1] C.W. Gardiner, P. Zoller, R.J. Ballagh and M.J. Davis,
Phys. Rev.Lett. 79, 1793 (1997).
60,000
[2] M.Anderson,J.R.Ensher,M.R.Matthews,C.E.Wieman
n and E.A. Cornell, Science 269, 198 (1995).
o
ati [3] K.B.Davis,M-O.Mewes.M.R.Andrews.N.J.vanDruten,
p
u D.S. Durfee, D.M. Kurn, and W. Ketterle, Phys. Rev.
c
Oc Lett. 75, 3969 (1995).
el [4] C.C. Bradley, C.A. Sackett, J.J. Tollet, and R. Hulet,
v
Le 0 Phys. Rev.Lett. 75, 1687 (1995).
0 0.1 [5] C.W. Gardiner and P. Zoller, cond-mat/9712002.
Time (sec) [6] M.Holland,J.Williams,K.Oakley,J.Cooper,Phys.Rev.
A 55, 3670 (1997)
Fig.4: Growthofnoncondensatelevels—thecondensatenum-
[7] E. Levich and V. Yakhot, Phys. Rev. B 15, 243 (1977);
ber is thealmost vertical black line.
J.Phys.A 11, 2237 (1978); D.W. Snoke and J.P. Wolfe,
Phys. Rev. B 39, 4030 (1989); H.T.C. Stoof, Phys. Rev.
In contrast to our work, in which explicit use is made of Lett. 66, 3148 (1991); Yu. M. Kagan, B. V. Svistunov
trap eigenfunctions, other descriptions [7–9] of conden- and G.V. Shlyapnikov, Sov. Phys JETP 75, 387 (1992);
sategrowtheitherdonottreatthecaseofatrappingpo- D.V.SemikozandI.I.Tkachev,Phys.Rev.Lett.74,3093
tential,orconsideronlythecasewheretrappedsituation (1995); H.T.C. Stoof, Phys. Rev.A 49, 3824 (1994).
is not essentially different from the untrapped situation. [8] Yu. Kagan and B. V. Svistunov, Phys. Rev. 79, 3331
Nevertheless, the equations we use have a strong con- (1997).
nection with those of Stoof [9], but their interpretation [9] H.T.C. Stoof, Phys. Rev.Lett. 78, 768 (1997).
4